topos theory

Contents

Idea

A geometric morphism $f:E\to F$ between toposes is a functor of the underlying categories that is consistent with the interpretation of $E$ and $F$ as generalized topological spaces.

If $F=\mathrm{Set}=\mathrm{Sh}\left(*\right)$ is the terminal sheaf topos, then $E\to \mathrm{Set}$ is essential if $E$ is a locally connected topos . In general, $f$ being essential is a necessary (but not sufficient) condition to ensure that $f$ behaves like a map of topological spaces whose fibers are locally connected: that it is a locally connected geometric morphism.

Definition

Definition

Given a geometric morphism $\left({f}^{*}⊣{f}_{*}\right):E\to F$, it is an essential geometric morphism if the inverse image functor ${f}^{*}$ has not only the right adjoint ${f}_{*}$, but also a left adjoint ${f}_{!}$:

$\left({f}_{!}⊣{f}^{*}⊣{f}_{*}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}E\stackrel{\stackrel{{f}_{!}}{\to }}{\stackrel{\stackrel{{f}^{*}}{←}}{\underset{{f}_{*}}{\to }}}F\phantom{\rule{thinmathspace}{0ex}}.$(f_! \dashv f^* \dashv f_*) \;\;\; : \;\;\; E \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} F \,.

A point of a topos $x:\mathrm{Set}\to E$ which is given by an essential geometric morphism is called an essential point of $E$.

Remark

There are various further conditions that can be imposed on a geometric morphism:

• If ${f}_{!}$ can be made into an $E$-indexed functor and ${f}^{*}$ satisfies some extra conditions, the geometric morphism $f$ is a locally connected geometric morphism (see there for details).

• If ${f}_{!}$ preserves finite products then $f$ is called connected surjective.

• If ${f}_{!}$ preserves finite products and moreover there is a further functor ${f}^{!}:F\to E$ which is right adjoint $\left({f}_{*}⊣{f}^{!}\right)$ and full and faithful, i.e. if we have a sequence of adjunctions

$\left({f}_{!}⊣{f}^{*}⊣{f}_{*}⊣{f}^{!}\right)$(f_! \dashv f^* \dashv f_* \dashv f^!)

with full and faithful ${f}^{!}$ then the geometric morphism $f$ is called local geometric morphism.

In this case in particular $F\stackrel{\stackrel{{p}_{*}}{←}}{\underset{{p}^{!}}{↪}}E$

is a geometric embedding and hence makes $F$ a subtopos of $E$.

Properties

Relation to morphisms of (co)sites

For $C$ and $D$ small categories write $\left[C,\mathrm{Set}\right]$ and $\left[D,\mathrm{Set}\right]$ for the corresponding copresheaf toposes. (If we think of the opposite categories ${C}^{\mathrm{op}}$ and ${D}^{\mathrm{op}}$ as sites equipped with the trivial coverage, then these are the corresponding sheaf toposes.)

Proposition

This construction extends to a 2-functor

$\left[-,\mathrm{Set}\right]:{\mathrm{Cat}}_{\mathrm{small}}^{\mathrm{co}}\to {\mathrm{Topos}}_{\mathrm{ess}}$[-,Set] : Cat_{small}^{co} \to Topos_{ess}

from the 2-category Cat${}_{\mathrm{small}}$ with 2-morphisms reversed) to the sub-2-category of Topos on essential geometric morphisms, where a functor $f:C\to D$ is sent to the essential geometric morphism

$\left({f}_{!}⊣{f}^{*}⊣{f}_{!}\right):\left[C,\mathrm{Set}\right]\stackrel{\stackrel{{f}_{!}:={\mathrm{Lan}}_{f}}{\to }}{\stackrel{\stackrel{{f}^{*}:=\left(-\right)\circ f}{←}}{\underset{{f}^{*}:={\mathrm{Ran}}_{f}}{\to }}}\left[D,\mathrm{Set}\right]\phantom{\rule{thinmathspace}{0ex}},$(f_! \dashv f^* \dashv f_!) : [C,Set] \stackrel{\overset{f_! := Lan_f}{\to}}{\stackrel{\overset{f^* := (-) \circ f}{\leftarrow}}{\underset{f^* := Ran_f}{\to}}} [D,Set] \,,

where ${\mathrm{Lan}}_{f}$ and ${\mathrm{Ran}}_{f}$ denote the left and right Kan extension along $f$, respectively.

Proposition

This 2-functor is a full and faithful 2-functor when restricted to Cauchy complete categories:

$\left[-,\mathrm{Set}\right]:{\mathrm{Cat}}_{\mathrm{CauchyComp}}^{\mathrm{co}}↪{\mathrm{Topos}}_{\mathrm{ess}}\phantom{\rule{thinmathspace}{0ex}}.$[-, Set] : Cat^co_{CauchyComp} \hookrightarrow Topos_{ess} \,.

For all small categories $C,D$ we have an equivalence of categories

$\mathrm{Func}\left(\overline{C},\overline{D}{\right)}^{\mathrm{op}}\simeq {\mathrm{Topos}}_{\mathrm{ess}}\left(\left[C,\mathrm{Set}\right],\left[D,\mathrm{Set}\right]\right)$Func(\overline{C},\overline{D})^{op} \simeq Topos_{ess}([C,Set], [D,Set])

between the opposite category of the functor category between the Cauchy completions of $C$ and $D$ and the the category of essential geometric morphisms between the copresheaf toposes and geometric transformations between them.

In particular, since every poset – when regarded as a category – is Cauchy complete, we have

Corollary

The 2-functor

$\left[-,\mathrm{Set}\right]:\mathrm{Poset}\to {\mathrm{Topos}}_{\mathrm{ess}}$[-,Set] : Poset \to Topos_{ess}
Remark

Sometimes it is useful to decompose this statement as follows.

There is a functor

$\mathrm{Alex}:\mathrm{Poset}\to \mathrm{Locale}$Alex : Poset \to Locale

which assigns to each poset a locale called its Alexandroff locale. By a theorem discussed there, a morphisms of locales $f:X\to Y$ is in the image of this functor precisely if its inverse image morphism ${f}^{*}\mathrm{Op}\left(Y\right)\to \mathrm{Op}\left(X\right)$ of frames has a left adjoint in the 2-category Locale.

Moreover, for any poset $P$ the sheaf topos over $\mathrm{Alex}P$ is naturally equivalent to $\left[P,\mathrm{Set}\right]$

$\left[-,\mathrm{Set}\right]\simeq \mathrm{Sh}\circ \mathrm{Alex}\phantom{\rule{thinmathspace}{0ex}}.$[-,Set] \simeq Sh \circ Alex \,.

With this, the fact that $\left[-,\mathrm{Set}\right]:\mathrm{Poset}\to \mathrm{Topos}$ hits precisely the essential geometric morphisms follows with the basic fact about localic reflection, which says that $\mathrm{Sh}:\mathrm{Locale}\to \mathrm{Topos}$ is a full and faithful 2-functor.

Some morphism calculus

Proposition

Let $f:E\to F$ be an essential geometric morphism.

For every $\varphi :X\to {f}^{*}{f}_{*}A$ in $E$ the diagram

$\begin{array}{ccc}X& \stackrel{\varphi }{\to }& {f}^{*}{f}_{*}A\\ ↓& & ↓\\ {f}^{*}{f}_{!}X& \stackrel{}{\to }& A\end{array}$\array{ X &\stackrel{\phi}{\to}& f^* f_* A \\ \downarrow && \downarrow \\ f^* f_! X &\stackrel{}{\to}& A }

commutes, where the vertical morphisms are unit and counit, respectively, and where the bottom horizontal morphism is the adjunct of $\varphi$ under the composite adjunction $\left({f}^{*}{f}_{!}⊣{f}^{*}{f}_{*}\right)$.

Proof

The morphism $\varphi :X\to {f}^{*}{f}_{*}A$ is the component of a natural transformation

$\begin{array}{ccccc}*& & \stackrel{X}{\to }& & E\\ {}^{A}↓& {⇓}^{\varphi }& & {}^{{f}^{*}}↗\\ E& \underset{{f}_{*}}{\to }& F\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ *&&\overset{X}{\to}&& E \\ {}^{\mathllap{A}}\downarrow &\Downarrow^\phi&& {}^{\mathllap{f^*}}\nearrow \\ E &\underset{f_*}{\to}&F } \,.

The composite $X\stackrel{\varphi }{\to }{f}^{*}{f}_{*}A\to A$ is the component of this composed with the counit ${f}^{*}{f}_{*}⇒\mathrm{Id}$.

We may insert the 2-identity given by the zig-zag law

$\cdots \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\begin{array}{ccccccccc}*& & \stackrel{X}{\to }& & E& & =& & E\\ {}^{A}↓& {⇓}^{\varphi }& & {}^{{f}^{*}}↗& ⇓& {↘}^{{f}_{!}}& ⇓& {↗}_{{f}^{*}}\\ E& \underset{{f}_{*}}{\to }& F& & =& & F\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\cdots \;\;\; = \;\;\; \array{ *&&\overset{X}{\to}&& E && = && E \\ {}^{\mathllap{A}}\downarrow &\Downarrow^\phi&& {}^{\mathllap{f^*}}\nearrow &\Downarrow& \searrow^{f_!} &\Downarrow& \nearrow_{\mathrlap{f^*}} \\ E &\underset{f_*}{\to}&F &&=&& F } \,.

Composing this with the counit ${f}^{*}{f}_{*}⇒\mathrm{Id}$ produces the transformation whose component is manifestly the morphism $X\to {f}^{*}{f}_{!}X\to A$.

Examples

Etale geometric morphisms

For any morphism $f:A\to B$ in a topos $E$, the induced geometric morphism $f:E/A\to E/B$ of overcategory toposes is essential.

For the case $B=*$ the terminal object, the geometric morphism

$\pi :E/A\to E$\pi : E/A \to E

is also called an etale geometric morphism.

Locally connected toposes

A locally connected topos $E$ is one where the global section geometric morphism $\Gamma :E\to \mathrm{Set}$ is essential.

$\left({f}_{!}⊣{f}^{*}⊣{f}_{*}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}E\stackrel{\stackrel{{\Pi }_{0}}{\to }}{\stackrel{\stackrel{\mathrm{LConst}}{←}}{\underset{\Gamma }{\to }}}\mathrm{Set}\phantom{\rule{thinmathspace}{0ex}}.$(f_! \dashv f^* \dashv f_*) \;\;\; : \;\;\; E \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} Set \,.

In this case, the functor ${\Gamma }_{!}={\Pi }_{0}:E\to \mathrm{Set}$ sends each object to its set of connected components. More on this situation is at homotopy groups in an (∞,1)-topos.

Note, though that if $p:E\to S$ is an arbitrary geometric morphism through which we regard $E$ as an $S$-topos, i.e. a topos “in the world of $S$,” the condition for $E$ to be locally connected as an $S$-topos is not just that $p$ is essential, but that the left adjoint ${p}_{!}$ can be made into an $S$-indexed functor (which is automatically true for ${p}^{*}$ and ${p}_{*}$). This is automatically the case for $\mathrm{Set}$-toposes (at least, when our foundation is material set theory—and if our foundation is structural set theory, then our large categories and functors all need to be assumed to be $\mathrm{Set}$-indexing anyway). For more see locally connected geometric morphism.

Tiny objects

The tiny objects of a presheaf topos $\left[C,\mathrm{Set}\right]$ are precisely the essential points $\mathrm{Set}\to \left[C,\mathrm{Set}\right]$. See tiny object for details.

References

The definition of geometric morphism appears before Lemma A.4.1.5 in

Connected surjective and local geometric morphisms are discussed in

Further refinements are in

• Bill Lawvere, Axiomatic cohesion Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (pdf)

Revised on September 17, 2011 08:33:17 by Toby Bartels (71.31.209.1)